Bank Directors Seminar, Coeur d'Alene, ID, September 15-17, 2019

SENSITIVITY TO MARKET RISK

Section 7.1

Duration-based price forecasts are generally precise when used with small rate changes (1 to 5 basis points). However, the accuracy of the forecasts decline when larger rates changes (especially 100 basis points or more) are involved. The reason for the declining accuracy of price forecasts relates to the non-linear relationship between prices and yields (a.k.a., convexity). Option-free financial instruments display positive convexity. When rates decline, a positively convexed instrument's price increases at an increasing rate. When rates rise, a positively convexed instrument's price decreases at a decreasing rate. Negative convexity causes the duration of a security to lengthen when rates rise and shorten when rates fall. Instruments that contain embedded options demonstrate negative convexity. When rates decline, a negatively convexed instrument's price increases at a decreasing rate. When rates rise, the price of a negatively convexed instrument will decline at an increasing rate. For example, the value of the treasury security changes relatively less in value in comparison to the sample mortgage security, which declines more significantly. However, as yields decrease, the treasury security gains value at an increasing rate, while the mortgage security gains only modestly. As interest rates decline, the likelihood increases that borrowers will refinance (exercise prepayment option). Therefore, the value of a mortgage security does not increase at the same rate or magnitude as a decline in interest rates. Effective duration and effective convexity are used to calculate the price sensitivity of bonds with embedded options. The calculations provide an approximate price change of a bond given a parallel yield curve shift. Measures of modified duration and convexity do not provide accurate calculations of price sensitivity for bonds with embedded options. Effective duration and convexity provide a more accurate view of price sensitivity since the measures allow for cash flows to change due to a change in yield. Formula: Effective Duration and Effective Convexity Convexity

V.. = Price if yield is decreased by Change Y V0= Initial price per $100 of par value

Assume: a three-year callable bond's current market value is $98.60 (V0); that interest rates are projected to change by 100 basis points (Y); that the price of this bond given a 100 basis point increase in rates is $96.75 (V+); and that the price of this bond given a 100 basis point decrease in rates is $99.98 (V..).

To calculate effective duration and convexity:

Effective Duration =

(99.98 — 96.75)/(2(98.60)(.01)) = 1.64

Effective Convexity =

96.75 + 99.98 — 2(98.60)÷(2(98.60)(.01))2 = -23.83

If we assume interest rates increase 100 basis points, the approximate price change due to effective duration is the following: Percentage Price Change = -Effective Duration x Yield Change Percentage Change in Price = -1.64 x .01 = -1.64% The approximate price change due to effective convexity is the following:

Yz x Effective Convexity x (Yield Change) 'A x -23.83 x (0.01)2 x 100 = -0.12%

Thus this bond's price would be expected to decrease by about 1.76 percent given a 100 bps rise in rates:

Effective Duration Effective Convexity

=

-1.64% -0.12%

=

-1.76%

Effective Duration = (V. - Vi.)/(2V0 x AY) Effective Convexity = (V++ VI - 2V0)/(2V0 x AY)2

Where, AY = Change in market interest rate used to calculate new values:

V+= Price if yield is increased by Change Y

Sensitivity to Market Risk (7/18)

7.1-24

RMS Manual of Examination Policies Federal Deposit Insurance Corporation

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